Not sure of the sources of the numbers in the Hallowell chart. But the source of the British numbers, when it comes to "equivalency" between US 7 and Brit 6, does not agree with what I find in my "Eley Shooter's Diary". And since Eley makes British shotshells, you'd think they'd be pretty accurate. They say 270 for Brit 6's. 299 for US 7's comes from most charts I've seen in the past. That's what I find in both a Lyman shotshell handbook and John Taylor's "Shotshells and Ballistics".
Of course as the Lyman book points out: "Pellets per ounce calculations based on soft, chilled shot. High antimony and/or plated shot is lighter. Actual count will vary depending on brand and lot." (Which I've confirmed by counting pellets in various factory loads, and my own reloads. Sometimes significantly.)
However, when it comes to Brit/US equivalents, I will buy Brit 4's = US 5's (both listed at 170/oz). And I won't quibble with Brit 5's = US 6 (220 to 225, respectively). But I don't think a 10% difference between Brit 6's and US 7's even meets the "close enough for govt work" standard.